Nicholas Witchell said royal officials had "desperately tried to help" the couple, but found it "very very difficult".
His comments come after the first three episodes of Harry and Meghan's Netflix six-part documentary aired to global audiences after months of waiting.
The initial episodes include criticism of the media and its intrusion into the couple's life, as well as Harry's experiences as a child.
Speaking on BBC News at Ten on Thursday, Witchell said Buckingham Palace would have been relieved that criticisms did not resurface over race in the royal family, saying: "His main target once again is the media, as it has been on so many occasions in the past. Press intrusion on a gross scale, that is a theme running through all three programmes."
He added: "But at times I think credibility is severely stretched."
He referred to a statement by Meghan in which she declared: "No matter what I did they were still going to find a way to destroy me."
Witchell said: "Really!? I’m not sure that claim would stand up to fair-minded and proper scrutiny.
"Royal officials who were assigned to them tried desperately to help them - they found it very very difficult. And yet it is the Sussexes who are convinced that they are the victims."
In the first three episodes of the series, titled Harry & Meghan, the couple address their issues with the tabloid press, with Harry suggesting that some members of the royal family questioned why Meghan should be protected any more than anyone else.
He said: "The direction from the Palace was don't say anything. But what people need to understand is, as far as a lot of the family were concerned, everything that she was being put through, they had been put through as well.
Watch: Harry and Meghan docu-series takes aim at tabloids
"So it was almost like a rite of passage, and some of the members of the family were like 'my wife had to go through that, so why should your girlfriend be treated any differently? Why should you get special treatment? Why should she be protected?'."
In the docuseries, Harry described the couple's relationship as a love story, saying: "She sacrificed everything she ever knew, the freedom that she had, to join me in my world. And then pretty soon after that I ended up sacrificing everything that I know to join her in her world."
He said it was his "duty" to "uncover this exploitation and bribery that happens within our media" and described the royal press pack as "essentially an extended PR arm of the Royal Family".
The couple also touched on a BBC interview that aired as part of the announcement of their engagement, with Meghan describing it as an "orchestrated reality show", saying: "We weren't allowed to tell our story... it was, you know, rehearsed."
Speaking of the press coverage she received when the couple first went public, Meghan said: “At that point, I was still very much believing what I was being told, which was ‘it will pass it will get better, it’s just what they do right at the very beginning’.
“This promise of ‘once you’re married, don’t worry, it’ll get better, once they get used to you it’ll get better, of course it’ll get better’.
“But truth be told, no matter how hard I tried, no matter how good I was, no matter what I did, they were still going to find a way to destroy me.”